The manufacturer of a motorcycle leases, the right to the brake system technology from motorcycle corporation the lease agreement calls for a payment of $20,000 at the beginning of each month for six years if the current interest rate is 6% compounding monthly bond value of payments

To calculate the present value (or bond value) of the lease payments, we'll treat the payment structure as an annuity due, since payments are made at the beginning of each month.

### Key Information:
- Monthly payment: $ P = 20,000 $
- Number of years: $ 6 $
- Monthly interest rate: $ i = \frac{6\%}{12} = 0.5\% = 0.005 $
- Total number of payments: $ n = 6 \times 12 = 72 $

### Formula for Present Value of an Annuity Due:
The present value $ PV $ of an annuity due is given by the formula:

\[
PV = P \times \left( \frac{1 - (1 + i)^{-n}}{i} \right) \times (1 + i)
\]

Where:
- $ P $ is the monthly payment
- $ i $ is the monthly interest rate
- $ n $ is the total number of payments

### Step-by-Step Calculation:

1. **Substitute the values**:
   \[
   PV = 20,000 \times \left( \frac{1 - (1 + 0.005)^{-72}}{0.005} \right) \times (1 + 0.005)
   \]

2. **Calculate** $ (1 + 0.005)^{-72} $:
   \[
   (1 + 0.005)^{-72} = 0.7047 \quad (\text{approx})
   \]

3. **Plug it into the formula**:
   \[
   PV = 20,000 \times \left( \frac{1 - 0.7047}{0.005} \right) \times 1.005
   \]

4. **Simplify the fraction**:
   \[
   \frac{1 - 0.7047}{0.005} = \frac{0.2953}{0.005} = 59.06
   \]

5. **Multiply by $ 20,000 $ and adjust for annuity due**:
   \[
   PV = 20,000 \times 59.06 \times 1.005 = 1,186,188 \quad (\text{approx})
   \]

### Final Result:
The present value of the lease payments is approximately **$1,186,188**.

Would you like more details on any of these steps, or have any questions?

Here are some related questions that you might find helpful:
1. How is an annuity due different from an ordinary annuity?
2. What is the effect of changing the interest rate on the present value?
3. How would the value change if payments were made at the end of the month?
4. What formula is used to calculate the future value of an annuity?
5. How do compound interest and simple interest differ in such scenarios?

**Tip:** When working with financial formulas, ensure that the interest rate period matches the payment frequency to avoid miscalculations.

The manufacturer of a motorcycle leases, the right to the brake system technology from motorcycle corporation. The lease agreement calls for a payment of $20,000 at the beginning of each month for six years. If the current interest rate is 6% compounding monthly, what is the bond value of payments?

Calculate the present value of monthly lease payments of $20,000 over six years with a 6% interest rate compounding monthly. This detailed guide explains annuity due calculations and includes key finance concepts such as present value, annuities, and interest rates.

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